Homogeneous solution assembled Turing structures with near zero strain semi-coherence interface

Turing structures typically emerge in reaction-diffusion processes far from thermodynamic equilibrium, involving at least two chemicals with different diffusion coefficients (inhibitors and activators) in the classic Turing systems. Constructing a Turing structure in homogeneous solutions is a large challenge because of the similar diffusion coefficients of most small molecule weight species. In this work, we show that Turing structure with near zero strain semi-coherence interfaces is constructed in homogeneous solutions subject to the diffusion kinetics. Experimental results combined with molecular dynamics and numerical simulations confirm the Turing structure in the spinel ferrite films. Furthermore, using the hard-soft acid-base theory, the design of coordination binding can improve the diffusion motion of molecules in homogeneous solutions, increasing the library of Turing structure designs, which provides a greater potential to develop advanced materials.

publication in Nature Comm after the appropriate revisions as mentioned below. The authors should include and address the following points in the revised manuscript clearly. 1. How can one confirm that the Turing structure is formed? Is ther any direct study/simulation etc. to support the claim. 2. How the images for Turing structure were developed from the SEM imagines is not clear. For example, for Supplementary Fig. 4. The authors should include details in the revised manuscript. 3. In Supplementary Fig. 4, the presence of α-Fe2O3 and ZnFe2O4 phases should be pointed clearly. 4. The information about the LED light used in this manuscript is missing. What is the wavelength of LED used? 5. How the authors calculated the 'molecular electrostatic potential' of the surfaces is not provided in Supplementary information. The authors should provide the details. 6. Supplementary Fig. 28f is not clear and difficult to understand. 7. What is the area or the volume of the electrode materials used for the study of the amount of O2 and H2 produced should be included. For example, in Fig.  Supplementary Fig. 30 b, c and d and other such plots in the whole manuscript in SI. 8. I do not understand the necessity to study the magnetic property of the material. Can the authors explain the same?
Reviewer #3 (Remarks to the Author): It is a nice piece of work on Interfacial engineering which would be useful for many energy applications. The work is interesting and can be published after incorporating these comments.
1. Though the role of turing structures is known, authors should include a paragraph on the mechanism and role of such turing structures in improving the charge recombination. 2. What is the role of FeCl3 and Fe(NO3)3 during the film formation process. 3. WHat is the reason of high output in the case of Turing interface film (Zn: Fe=1:3) 4. An explanation to be added to why charge transfer is no existent in bulk of Zn ferrite. 5. Can this be applied to perovskite solar cells as interfacial properties play a major role in the same. 6. What is the surface roughness of the film and did it have any influence on the efficiency. 7. Do the fabrication of these films have a role in deciding the turing structures. 8. What is the thickness of the films. 9. Similar reports should be compared in the discussion. In the manuscript "Homogeneous solution assembled Turing structures with near zero strain semi-coherence interface," Zhang et al. reported that the Turing structure can be realized for an inorganic semiconductor thin film system by re-coordination in homogeneous solutions. The authors concluded that the Turing interface can enhance charge separation efficiency.
The experimental data are technically sound, with various measurements shown in Supplementary Information (SI). Nevertheless, it must be said that this paper has serious flaws.

Response:
We are very grateful for the reviewer. The review comments are very constructive to further improve the quality of the manuscript. We have addressed the comments point-by-point and made the corresponding changes accordingly in the revised manuscript. We hope our revision with new experimental results and the expanded theoretical simulations has convinced the reviewer of the significance and the originality of our work.
The authors claim that the pattern in Fig. 1c is the Turing pattern. However, no clear evidence supports this claim. The authors must provide evidence of the Turing pattern in sufficient quality and quantity. It is well known in mathematical biology that one cannot claim the Turing pattern just because it looks like the Turing pattern. If the authors insist on claiming that their interface film is the Turing pattern, they should at least show that the same pattern can be derived from Turing's reaction-diffusion equation.

Response:
We thank the reviewer for insightful and constructive suggestions. As the reviewer pointed out, one cannot claim the Turing pattern just because it looks like the Turing pattern.
According to the suggestion, we conducted the experiments and simulations, including two-dimensional diffusion-ordered nuclear magnetic resonance spectroscopy (2D DOSY) (Fig.   3a), molecular dynamics (MD) simulation  and numerical simulation ( Fig. 3h and Supplementary Fig. 19). We will analyze these results in detail below. , and the initial data is in line with the equation 14.
Turing structures typically emerge in reaction-diffusion processes far from thermodynamic equilibrium, involving at least two chemicals with different diffusion coefficients (inhibitors and activators) [r1, r2]. Alan Turing mathematically showed that the stable state can destabilize under certain conditions and spontaneously generate spatially stationary patterns in a reaction-diffusion system, which is particularly well known in mathematical biology. Thus, the Turing pattern can be described by dimensionless reaction-diffusion equations combined with our chemical system: respectively. The generation of the Turing pattern corresponds to the coupling of a nonlinear reaction kinetic process and a special diffusion process, which will be unstable due to the different diffusion velocities of the two factors.
First, we verified the activators and inhibitors with big difference in diffusion coefficient in homogeneous solution by the 2D DOSY and MD simulation. As shown in Fig. 3a, it should be noted that the diffusion coefficient of (CH 3 COO)was 4 times higher than the diffusion coefficient  (Fig. 3c), meaning that the dynamic process of Zn 2+ is slow. While the movement of Zn 2+ and Fe 3+ was not affected by NO 3 - (Fig. 3d) , we expect to see the formation of patterns. Furthermore, the given parameter value according to our chemical system will fall into the Turing space, giving rise to diff erent spatial patterns with respect to time, as shown in Fig. 3h. Turing patterns can still be obtained by changing key parameters such as diffusion coefficient difference ( ) and initial conditions (Supplementary Fig. 19 Applying the scaling to system: The Jacobian matrix at the positive equilibrium is shown below: We can derive from the linear system: The characteristic function of equation 5 is as follows: The condition for a spatial mode defined by to be unstable and thus to form a pattern in equation 9. ( ) is a function of ( 2 ), and ( 2 ) is closely related with . Therefore, ( = ) determines the range of ( ) and the stability of the diffusion system. Next, we use numerical simulation [r4] to verify whether the experimental system is unstable. First, we simplify equation 4 to the following form: For the two-dimensional approximations, we use a uniform subdivision of the square by finite difference method.
[r1] Z. Tan respectively. In these systems, there are two chemical reactants that can not only interact, but also diffuse alone. Hence, the generation of the Turing pattern corresponds to the coupling of a nonlinear reaction kinetic process and a special diffusion process, which will be unstable due to the different diffusion velocities of the two factors. To demonstrate the instability caused by this diffusion, we continue to analyze the difference between the diffusion coefficients of the two substances by means of a combination of experiments and theory. As shown in Fig. 3a, it should be noted that the diffusion coefficient of (CH 3 COO)was 4 times higher than the diffusion coefficient of [C 5 H 7 O 2 ]in deuterated methanol for the 2D DOSY test due to the difference in molecular weight.  , we expect to see the formation of patterns.
Furthermore, the given parameter value according to our chemical system will fall into the Turing space, giving rise to diff erent spatial patterns with respect to time, as shown in Fig. 3h. Turing patterns can still be obtained by changing key parameters such as diffusion coefficient difference ( ) and initial conditions ( Supplementary Fig. 19). The consistency between experiments and simulations verified that the formation of the architecture is closely related to Turing"s theory. and activator (prey), respectively. This type of model takes the form 23 : Applying the scaling to system: where ⃗ is space vector.
The Jacobian matrix at the positive equilibrium is shown below: We can derive from the linear system: The characteristic function of equation 5 is as follows: The condition for a spatial mode defined by to be unstable and thus to form a pattern in equation 9.
( ) is a function of ( 2 ), and ( 2 ) is closely related with . Therefore, ( = ) determines the range of ( ) and the stability of the diffusion system. Next, we use numerical simulation 24 to verify whether the experimental system is unstable. First, we simplify equation 4 to the following form: , and the initial data is in line with the equation 14." In addition to this, the following statements are overstatements because there is no enough evidence to support their proposal: "we ... proposed a new concept of a Turing interface at the atomic level" "a new concept of a Turing interface at the atomic level was presented for the first time," Response: Thanks for the reviewer"s valuable comments. We have modified the above statement as follows: "which was named Turing interface at the atomic level" "the Turing interface at the atomic level was revealed." From the viewpoint of readability, the manuscript is not well written. They cite SI too often, preventing readers from understanding the essence of the paper. The main text itself should powerfully state the author's views.
For these reasons, I cannot recommend this manuscript for publication in Nature Communications.

Response:
We thank the reviewer to point this out. According to the suggestion, we have moved the evidence from the Supporting Information to the main text and adjusted some figures in the Supporting Information, which can strongly state our views.
(1). More information about interfacial strain was added to the main text, as shown in Figure 5i and j (2). A description of the thickness of the film was added to the main text as follows: "In addition, the optimized film thicknesses were kept at 300 nm, 300 nm and 3.2 μm for the Turing interface films, non-Turing dual-phase interface films and conventional dual-phase interface films ( Supplementary Fig. 35b)." (3). A description of the surface roughness of films and the influence of fabrication process was added to the main text as follows: "To exclude the influence of the surface structure of the film, we conducted a surface roughness measurement. According to the AFM micrographs (2D and 3D) in Supplementary Fig.   46 and 47, there is little difference between the Turing structure film and the non-Turing structure film, and the surface roughness is distributed at 5-8 nm. Therefore, we can be sure that the same preparation process has little effect on the surface roughness of the film. Moreover, there is no strong dependence between its performance and surface roughness. We confirm that the enhancement in the efficiency of the thin film is attributed to the formation of the bulk Turing interface. In order to further verify that the formation of the Turing structure is determined by the intrinsic property of the solution rather than an artifact that can appear due to the fabrication process, we used the drop coating and spin coating technique to prepare high-performance Turing    respectively. The generation of the Turing pattern corresponds to the coupling of a nonlinear reaction kinetic process and a special diffusion process, which will be unstable due to the different diffusion velocities of the two factors.
First, we verified the activators and inhibitors with big difference in diffusion coefficient in homogeneous solution by the 2D DOSY and MD simulation. As shown in Fig. 3a, it should be noted that the diffusion coefficient of (CH 3 COO)was 4 times higher than the diffusion coefficient , we expect to see the formation of patterns. Furthermore, the given parameter value according to our chemical system will fall into the Turing space, giving rise to diff erent spatial patterns with respect to time, as shown in Fig. 3h. Turing patterns can still be obtained by changing key parameters such as diffusion coefficient difference ( ) and initial conditions ( Supplementary Fig.   19). The consistency between experiments and simulations verified that the formation of the architecture is closely related to Turing"s theory.
where ⃗ is space vector.
The Jacobian matrix at the positive equilibrium is shown below: We can derive from the linear system: The characteristic function of equation 5 is as follows: The condition for a spatial mode defined by to be unstable and thus to form a pattern in equation 9.
( ) is a function of ( 2 ), and ( 2 ) is closely related with . Therefore, ( = ) determines the range of ( ) and the stability of the diffusion system. Next, we use numerical simulation [r4] to verify whether the experimental system is unstable. First, we simplify equation 4 to the following form: When = 1, = 5, = 10, ⋯ , = 1000, the simulated Turing pattern can be obtained.
[r1] Z. [r4] G.P. Hu, Z.S. Feng. Turing instability and pattern formation in a strongly coupled diff usive predator-prey system. Int. J. Bifurcation and Chaos. 30, 2030020-1-15 (2020). respectively. In these systems, there are two chemical reactants that can not only interact, but also diffuse alone. Hence, the generation of the Turing pattern corresponds to the coupling of a nonlinear reaction kinetic process and a special diffusion process, which will be unstable due to the different diffusion velocities of the two factors. To demonstrate the instability caused by this diffusion, we continue to analyze the difference between the diffusion coefficients of the two substances by means of a combination of experiments and theory. As shown in Fig. 3a, it should be noted that the diffusion coefficient of (CH 3 COO)was 4 times higher than the diffusion coefficient of [C 5 H 7 O 2 ]in deuterated methanol for the 2D DOSY test due to the difference in molecular weight.   (Fig. 3c), meaning that the dynamic process of Zn 2+ is slow. While the movement of Zn 2+ and Fe 3+ was not affected by NO 3 - (Fig. 3d).
The above experiments and MD results provide solid evidence that high molecular weight organic anions play a key role in homogeneous solution for formation of Turing patterns. Accordingly, we , we expect to see the formation of patterns.
Furthermore, the given parameter value according to our chemical system will fall into the Turing space, giving rise to diff erent spatial patterns with respect to time, as shown in Fig. 3h. Turing patterns can still be obtained by changing key parameters such as diffusion coefficient difference ( ) and initial conditions ( Supplementary Fig. 19). The consistency between experiments and simulations verified that the formation of the architecture is closely related to Turing"s theory.
The Jacobian matrix at the positive equilibrium is shown below: We can derive from the linear system: The characteristic function of equation 5 is as follows: The condition for a spatial mode defined by to be unstable and thus to form a pattern in equation 9.
( ) is a function of ( 2 ), and ( 2 ) is closely related with . Therefore, ( = ) determines the range of ( ) and the stability of the diffusion system. Next, we use numerical simulation 24 to verify whether the experimental system is unstable. First, we simplify equation 4 to the following form: , and the initial data is in line with the equation 14." "In order to further verify that the formation of Turing structure is determined by intrinsic property of the solution rather than an artifact that can appear due to the fabrication process, we used the drop coating and spin coating technique to prepare high-performance Turing structure films. The above conclusions are furtherly summarized from Supplementary Fig. 48. Once the difference of diffusion coefficient of related substances in homogeneous solution is modulated, Turing structure films can be obtained by either drop coating or spin coating process. Furthermore, the Turing structure films performances were found to be better than that of non-Turing structure films. Nevertheless, there is a gap between the properties of the films prepared by the above process and the films prepared by spray pyrolysis process. It is mainly due to the characteristics of spin coating and drop coating process, and we have not optimized the technology. For the drop coating process, although Turing structure film can also be prepared, the film area is usually small and the thickness is not easy to control. For the spin coating technique, the viscosity of the Response: We thank the reviewer for the good suggestion. The calculation description of the molecular electrostatic potential is added to the computational details of the revised manuscript.
"Gauss View can create a surface where the color is determined by the values of a second property, which implies that we can map the values of one property on an isosurface of a different property.
The cubegen (a module of Gaussian software) was first employed to calculate the electron density isosurface (0.001 e/bohr 3 ). Subsequently, the electrostatic potential value of the point on the isosurface was calculated and displayed on the electron isosurface by color. Then, according to the electrostatic potential value of different molecules, the electron density surface was drawn." 6. Supplementary Fig. 28f is not clear and difficult to understand.
Response: We appreciate the reviewer"s valuable comments that help to improve the quality of our work. We are quite sorry for the unclear display of this picture. We redraw it as shown in and H 2 produced should be included. For example, in Fig. Supplementary Fig. 30 b, c and d and other such plots in the whole manuscript in SI.
Response: Thanks for the reviewer"s suggestion. We apologize that in our previous manuscript the related description was not explicit. We have revised the caption of Supplementary Fig. 30 ( Supplementary Fig. 28 in our previous Supplementary Information), Supplementary Fig. 32 ( Supplementary Fig. 30 in our previous Supplementary Information) and the description in the manuscript as follows: "The Turing interface film (Zn: Fe=1:3) with an eff ective area of 1 cm 2 showed excellent stability ( Supplementary Fig. 30c)  It is a nice piece of work on Interfacial engineering which would be useful for many energy applications. The work is interesting and can be published after incorporating these comments.
Response: We sincerely thank the reviewer"s comments. The proposed suggestions are valuable and helpful for improving our work We have carefully revised the manuscript and replied to the comments point-by-point shown below.
1. Though the role of turing structures is known, authors should include a paragraph on the mechanism and role of such turing structures in improving the charge recombination.
Response: We appreciate the reviewer"s valuable comments very much. We are also sorry not to provide a clear description in the previous manuscript. In revised manuscript, we have made more profound and comprehensive discussions about the charge separation enhancement mechanism of the Turing structure film.
"Photoelectrochemical measurement and charge separation enhancement mechanism.
Having identified the essential roles of the interface structure, we propose an interface transport mechanism that shows a large difference in the separation efficiency of photogenerated carriers. In the photoelectrode film, electron-hole pairs are generated under light, followed by different charge transport and trapping processes under bias. In the Turing interface film (Fig. 6a), the generated electron-hole pairs separate effectively at the interface via an enhanced interface built-in electric field. In the conventional dual-phase interface film (Fig. 6b), a built-in electric field at the interface is weakened due to the existence of the incoherent interface or semi-coherent interface.
Therefore, the electron and hole pairs cannot be effectively separated at the interface, which increases the probability of recombination within the particle. In the case of a non-Turing dual-phase interface film (Fig. 6c), the built-in electric field at the interface is also weakened, and the separated electrons and holes are recombined at the interface due to the mutual coating of the dual-phases. Above was further evaluated under AM 1.5 G irradiation with a standard three-electrode system in 1 M NaOH solution."  3. What is the reason of high output in the case of Turing interface film (Zn: Fe=1:3).
Response: We are greatly grateful to the reviewer"s nice question and it is very useful for improving the quality of this work. Turing interface film (Zn: Fe=1:3) exhibits excellent photoelectrochemical water splitting and phenol oxidation due to its higher charge separation efficiency. More importantly, the unique interface features formed by the Turing structure.
According to the analysis of abound precise atomic interfaces in the film as shown in Fig. 5, semi-coherent interface with an interface strain close to zero is formed between the dual phases.
Therefore, it is essential to consider the interface state in understanding the equilibrium energy band diagram as shown in Supplementary Fig. 38 (Supplementary Fig. 36 in our previous Supplementary Information). Thus, a strong interface built-in electric field is formed in the Turing interface film (Zn: Fe=1:3), which promotes the separation of photogenerated electron-hole pairs. 4. An explanation to be added to why charge transfer is non existent in bulk of Zn ferrite.
Response: It is a very good question, and we are sorry that charge transfer process was not stated clearly in the previous manuscript. Generally speaking, transient photovoltage (TPV) is an effective method to study the dynamics of photogenerated carriers [r1, r2]. Photo-generated charge separation process includes drift and diffusion: the drift process refers to the fast separation process (<10 -5 s) that occurs within the particles; the main diffusion process is the charge transfer between particles over a long period of time (>10 -4 s). Thus, the photovoltage response includes two parts: rising and decay. The rising part of the photovoltage physically corresponds to the increase in the electron concentration of the conductive substrate of the electrode. Caused by the diffusion of photogenerated electrons to the substrate, the drop in photovoltage mainly corresponds to the recombination process of electrons leaving the conductive substrate. Therefore, charge transfer was tough in the bulk for ZnFe 2 O 4 due to the fast recombination of photogenerated electrons and holes compared with the Turing interface film (Zn: Fe=1:3) (Fig. 6g). In the revised manuscript, the following text has been added:  (Fig. Ra). After addition, the film presented a dense structure (Fig.   Rb), and an obvious phase dispersion structure appears (Fig. Rc) when it continues to increase.
Furthermore, XRD revealed that there was not much difference in the phases. Next, we assembled the device to evaluate photovoltaic parameters as shown in Fig. Re- 6. What is the surface roughness of the film and did it have any influence on the efficiency.
Response: We are greatly grateful to the reviewer for the nice question. We agree with the comment that the surface roughness of the film may influence the efficiency. In this regard, the surface roughness of the film will be analyzed in detail. In the previous manuscript, we measured "To exclude the influence of the surface structure of the film, we conducted a surface roughness measurement. According to the AFM micrographs (2D and 3D) in Supplementary Fig.   46 and 47, there is little difference between the Turing structure film and the non-Turing structure film, and the surface roughness is distributed at 5-8 nm. Therefore, we can be sure that the same preparation process has little effect on the surface roughness of the film. Moreover, there is no strong dependence between its performance and surface roughness. We confirm that the enhancement in the efficiency of the thin film is attributed to the formation of the bulk Turing interface." 7. Do the fabrication of these films have a role in deciding the turing structures.
Response: We are greatly grateful to the reviewer for the nice question and careful inspection. We have attempted to fabricate the films by different techniques and collect the SEM images which was found similar irrespective to the fabrication process. Therefore, the Turing structure was an inherent feature of the homogeneous solution, which is not limited by the fabrication process. The above conclusions are furtherly summarized from Supplementary Fig. 48. Once the difference of diffusion coefficient of related substances in homogeneous solution is modulated, Turing structure films can be obtained by either drop coating or spin coating process. Furthermore, the Turing structure films exhibited better performances than non-Turing structure films. Nevertheless, there is a gap between the properties of the films prepared by the above process and the films prepared by spray pyrolysis process. It is mainly due to the characteristics of spin coating and drop coating process, and we have not optimized the technology. For the drop coating process, although Turing structure film can also be prepared, the film area is usually small and the thickness is not easy to control. For the spin coating technique, the viscosity of the solution needs to be considered. If the process is optimized, a film with performance comparable to that of the spray pyrolysis film can be obtained. Accordingly, here we try to improve the spin coating technique and give researchers more inspiration. By comparing the droplet evolution of mixed solutions composed of iron sources and Zn(CH 3 COO) 2 ( Supplementary Fig. 49 "In order to further verify that the formation of Turing structure is determined by intrinsic property of the solution rather than an artifact that can appear due to the fabrication process, we used the drop coating and spin coating technique to prepare high-performance Turing structure films. The above conclusions are furtherly summarized from Supplementary Fig. 48. Once the difference of diffusion coefficient of related substances in homogeneous solution is modulated, Turing structure films can be obtained by either drop coating or spin coating process. Furthermore, the Turing structure films performances were found to be better than that of non-Turing structure films. Nevertheless, there is a gap between the properties of the films prepared by the above process and the films prepared by spray pyrolysis process. It is mainly due to the characteristics of spin coating and drop coating process, and we have not optimized the technology. For the drop coating process, although Turing structure film can also be prepared, the film area is usually small and the thickness is not easy to control. For the spin coating technique, the viscosity of the solution needs to be considered. If the process is optimized, a film with performance comparable to that of the spray pyrolysis film can be obtained. Accordingly, here we try to improve the spin coating technique and give researchers more inspiration. By comparing the droplet evolution of mixed solutions composed of iron sources and Zn(CH 3 COO) 2 ( Supplementary Fig. 49), we can draw the following conclusion: Fe(NO 3 ) 3 and FeCl 3 in four solvents show a diffusion coefficient difference close to 1, which does not appear regular spots or strips structure. Fe[C 5 H 7 O 2 ] 3 is used as a source of iron, and ethylene glycol or acetylacetone as a solvent, which will not form a Turing structure. While it exhibits a regular structure in DMF solution. Simultaneously, introducing small amount of surfactant into the methanol solution does not affect the diffusion coefficient of the ions, so as to adjust the viscosity of the solution, it will be easier to spin coating into a high-quality film. This further confirms that the Turing structures of film is the intrinsic property of the solution." 8. What is the thickness of the films.
Response: We are greatly grateful to the reviewer for the nice question. We optimized the thickness of the film to the best in the experiment, the thickness of the film was maintained at 300 nm as shown in Supplementary Fig. 18 (Supplementary Fig. 5 in our previous Supplementary   Information), but we are sorry that we have less analysis of the film thickness. Here, we study that inorganic Turing films have high photogenerated charge separation efficiency. In order to highlight the advantages of this kind of films, we prepared non-Turing dual-phase interface film and conventional dual-phase interface film. Furtherly, we add the thickness of conventional dual-phase interface film in Supplementary Fig. 5 (Supplementary Fig. 7

in our previous
Supplementary Information) and non-Turing dual-phase interface film in Supplementary Fig. 6 (Supplementary Fig. 8  Turing-type polyamide membranes for water purification, and these membranes exhibit excellent water-salt separation performance. Zhang 24 reported a cation exchange approach in the heterogeneous solvent of diethylenetriamine and deionized water to produce Turing-type Ag 2 Se on CoSe 2 nanobelts relied on diffusion-driven instability, which is highly effective in catalyzing the oxygen evolution reaction (OER) in alkaline electrolytes with an 84.5% anodic energy efficiency.
It should be noted that the above case studies the related potential of the Turing structure in heterogeneous solution." (ID: NCOMMS-21-39616A) General response: We sincerely thank the editor, editorial staff and all reviewers for their critical comments that we have based on to improve the quality of our manuscript. The manuscript has been modified point-by-point after addressing all the suggestions as listed below.
(Our response is given in blue, some key sentences are highlighted in yellow and the corrections in the revised manuscript are shown in red) Reviewer #1 (Remarks to the Author): The authors replied to the referees' comments point-by-point and gave appropriate answers. In the revised manuscript and the reply, the authors have largely improved the theoretical support on the evidence of the Turing pattern formation with the numerical simulation based on the predator-prey model. The difference in the diffusion constant between Fe and Zn can be a reasonable condition of Turing pattern formation. They obtained numerical results that bear some resemblance to the observed patterns. Although it is not strong evidence of the Turing pattern, it may include the truth.
We cannot judge the correctness at this state, but I think it is constructive for the related fields to propose one possibility and stimulate further investigations on this topic. In this sense, I think the manuscript can be considered for publication in Nature Communications.
Response: We are very grateful for the reviewer. The reviewer's comments are very constructive and insightful to further improve the quality of the manuscript. We have made a point-by-point responses to address the reviewer's concerns.
In a reaction-diffusion system, the stable state can destabilize under certain conditions and spontaneously create space stable pattern. Therefore, the generation of the Turing pattern corresponds to the coupling of a nonlinear reaction kinetic process and a diffusion process. From the understanding of the mathematical mechanism, the stable constant equilibrium state of the ordinary differential systems undergoes a stability reversal after the addition of diffusion.
Therefore, we also believe that in the future, more systems will be able to design Turing structure from the basic mathematical point of view, so as to promote the development of this field.
However, it is desirable to add reasonable explanations on the following points.
(1) There are various types of models that generate the Turing patterns. Why is the current predator-prey model the most plausible model?
Response: We thank for the reviewer's meaningful comments.
As research progresses, various types of models that can generate Turing patterns are developed, including Lotka-Volterra (predator-prey), Gierer-Meinhardt, Lengyel-Epstein, Thomas and so on [r1]. In the early years, Turing's reaction-diffusion model for pattern formation was attracted more attention from theoretical biology and applied mathematics [r2]. Alan Turing expressed this mechanism of pattern formation in terms of simultaneous differential equations of the form: (1) auto-catalysis and cross-catalysis must exist between an activator and an inhibitor; (2) the diffusion of the inhibitor must be much faster than that of the activator [r3-r5].
In our experiment, substance A (Fe) and substance B (Zn) with large difference in diffusion coefficient correspond to inhibitor and activator, respectively. The two substances are simultaneously affected by cross-diffusion according to the hard-soft acid-base theory. At the same time, the reacted substance concentrations are all positive values. These features fit best with classical predator-prey models. The prey acts as an activator, seeking to reproduce and increase their numbers, while the predator acts as an inhibitor, keeping populations in check. More importantly, when conducting relevant research in the fields of economy, physics and chemistry, scholars often select predator-prey model and convert the differential equation into difference [r7] G.P. Hu, Z.S. Feng. Turing instability and pattern formation in a strongly coupled diff usive predator-prey system. Int. J. Bifurcation and Chaos. 30, 2030020-1-15 (2020).
The related discussion has been added in the revised manuscript. Although other models can also be numerically simulated, they are difficult to match the complex conditions of chemical reactions." (2) There are various parameters in the present model. What are their relationships to the experiment?
Response: Thanks for the reviewer's valuable comments.
In the derivation part of the predator-prey model in the Supplementary Information, we verified that the occurrence of Turing instability is closely related to the difference of the diffusion coefficients of the two substances, so the difference of the diffusion coefficients of the substances is a key parameter connecting the experiment and the model ( = ) according to the according to the following predator-prey model: